Is randomness anti-logical?
I think the answer here lies in that the dictionary definition of randomness you have is not the one used in statistics, ML, or mathematics. We define randomness to mean there exists a distribution with generally greater than 0 uncertainty.  

Depending on who you talk to, we live in a random universe (the way we define quantum mechanics depends on a wave function (essentially a probability distribution)  

So why if a sequence is drawn from a distribution is it illogical? First, even as humans we can make a strong argument that what we say is random. I mean we speak to convey some form of message or context, but there exists multiple ways to deliver this, but we choose a single one. Our brains inherently model $p(\vec w|c)$ where $\vec w$ is the sequence and $c$ is our context or message we want to convey.  

Takeaway: Generating a sequence in an ergodic or uniform manner would be illogical, but that is not what is being modeled or done in practice. Normally its drawn from some complex distribution. 

Sidenote: My above claim could make it seem that being uniformly random implicates something illogical, and I want to emphasize that is not the case. It is domain to domain, sometimes that is the most logical solution, just in the case of sentence generation it normally isnt. I would define a logical algorithm as one that given the information at hand acts in a sensible manner towards achieving some goal, and so if something purely random does that, I don't see the problem.